Computational Modeling and Verification of Signaling...
Transcript of Computational Modeling and Verification of Signaling...
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Haijun Gong*, Paolo Zuliani*, Anvesh Komuravelli*,
James R. Faeder#, Edmund M. Clarke*
*Computer Science Department, Carnegie Mellon University#School of Medicine, University of Pittsburgh
Computational Modeling and Verification of Signaling
Pathways in Cancer
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07/16/0907/16/0907/16/0907/16/0907/16/0907/16/09
The Hallmarks of Cancer
D. Hanahan and R. A. WeinbergCell, Vol. 100, 57–70, January 7, 2000
“Six essential alterations in cell physiology that collectively dictate malignant growth.”
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07/16/0907/16/0907/16/0907/16/0907/16/0907/16/09
The Hallmarks of Cancer
D. Hanahan and R. A. WeinbergCell, Vol. 100, 57–70, January 7, 2000
All cancers share the six alterations.
The way the alterations are acquired varies, both mechanistically and chronologically.
Can we formalize the acquisition processes?
Is there an “integrated circuit of the cell”?
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07/16/0907/16/0907/16/0907/16/0907/16/0907/16/09
The Cell Integrated Circuit (?)
D. Hanahan and R. A. WeinbergCell, Vol. 100, 57–70, January 7, 2000
Completed by 2020?
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2010: the “integrated circuit of the cell” still not in sight …
But computational models can compare qualitatively well with experiments.
We use the BioNetGen language (http://bionetgen.org) to describe signaling pathways important in many cancers:
We focus on the HMGB1 protein and the p53, NFkB, RAS and Rb signaling pathways
We use statistical model checking to formally verify behavioral properties expressed in temporal logic:
Can express quantitative properties of systems
Scalable, can deal with large models07/16/0907/16/0907/16/0907/16/0907/16/0907/16/09
This Work
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Signaling Pathways
p53-MDM2 and PI3K-AKT pathways
RAS-ERK pathway
Rb-E2F pathway
NFkB pathway
HMGB1
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In resting cells IkB is found only in the cytoplasm, bound to NFkB
HMGB1 can break the complex and liberate NFkB
NFkB enters the nucleus …
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The Rb-E2F pathway is important in the cell cycle
It regulates the G1-S transition
Rb keeps E2F in a complex
HMGB1 can break it and liberate E2F
E2F activates the transcription of CyclinE …
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44 molecular species82 reactions
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BioNetGen.org
Rule-based modeling for biochemical systems
Ordinary Differential Equations and Stochastic simulation (Gillespie’s algorithm)
Example: AKT has a component named d which can be labeled as U (unphosphorylated) or p (phosphorylated)
begin species begin parameters
AKT(d~U) 1e5 k 1.2e-7
AKT(d~p) 0 d 1.2e-2
end species end parameters
Faeder JR, Blinov ML, Hlavacek WS Rule-Based Modeling of Biochemical Systems with BioNetGen. In Methods in Molecular Biology: Systems Biology, (2009).
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BioNetGen.org
PIP3 can phosphorylate AKT, and dephosphorylation of AKT
begin reaction_rules
PIP(c~p) + AKT(d~U) → PIP(c~p) + AKT(d~p) k
AKT(d~p) → AKT(d~U) d
end reaction_rules
The corresponding ODE (assuming AKT+AKTp=const) is:
AKTp(t)' = k∙PIP3(t)∙AKT(t) – d∙AKTp(t)
The propensity functions for Gillespie’s algorithm are:
k∙[PIP(c~p)]∙[AKT(d~U)]
d∙[AKT(d~p)]
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Verification of BioNetGen Models
Temporal properties over the model’s stochastic evolution
For example: “does AKTp reach 4,000 within 20 minutes, with probability at least 0.99?”
In our formalism, we write:
P≥0.99 (F20 (AKTp ≥ 4,000))
For a property Ф and a fixed 0<θ<1, we ask whether
P≥θ (Ф) or P<θ (Ф)
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A biased coin (Bernoulli random variable):
Prob (Head) = p Prob (Tail) = 1-p
p is unknown
Question: Is p ≥ θ ? (for a fixed 0<θ<1)
A solution: flip the coin a number of times, collect the outcomes, and use: Statistical hypothesis testing: returns yes/no Statistical estimation: returns “p in (a,b)” (and compare a with θ)
Equivalently
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Statistical Model Checking
Key idea Suppose system behavior w.r.t. a (fixed) property Ф can be
modeled by a Bernoulli random variable of parameter p:
System satisfies Ф with (unknown) probability p
Question: P≥θ (Ф)? (for a fixed 0<θ<1)
Draw a sample of system simulations and use: Statistical hypothesis testing: Null vs. Alternative hypothesis
Statistical estimation: returns “p in (a,b)” (and compare a with θ)
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Motivation
Pros: Simulation is feasible for many systems
Often easier to simulate a complex system than to build the transition relation for it
Easier to parallelize
Cons: answers may be wrong
But error probability can be bounded
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Statistical Model Checking of biochemical models: M╞═ P≥θ(Φ)?
Model MStochastic simulation
BioNetGenStatistical Model Checker
Temporal property Φ
Formula monitor
M╞═ P≥θ (Φ)
Statistical Test
M╞═ P≥θ (Φ)
Our Approach
Error probability
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a sample of Bernoulli random variables
Prior probabilities P(H0), P(H1) strictly positive, sum to 1
Posterior probability (Bayes Theorem [1763])
for P(X) > 0
Ratio of Posterior Probabilities:
Bayes Factor
Sequential Bayesian Statistical MC - I
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Recall the Bayes factor
Jeffreys’ [1960s] suggested the Bayes factor as a statistic: For fixed sample sizes
For example, a Bayes factor greater than 100 “strongly supports” H0
We introduce a sequential version of Jeffrey’s test
Fix threshold T ≥ 1 and prior probability. Continue sampling until
Bayes Factor > T: Accept H0
Bayes Factor < 1/T: Reject H0
Sequential Bayesian Statistical MC - II
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Require: Property P≥θ(Φ), Threshold T ≥ 1, Prior density gn := 0 {number of traces drawn so far}x := 0 {number of traces satisfying Φ so far}repeat
σ := draw a sample trace from BioNetGen (iid)n := n + 1if σ Φ then
x := x + 1endifB := BayesFactor(n, x, θ, g)
until (B > T v B < 1/T )if (B > T ) then
return “H0 accepted”else
return “H0 rejected”endif
Sequential Bayesian Statistical MC - III
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Correctness
Theorem (Termination). The Sequential Bayesian Statistical MC algorithm terminates with probability one.
Theorem (Error bounds). When the Bayesian algorithm – using threshold T – stops, the following holds:
Prob (“accept H0” | H1) ≤ 1/T
Prob (“reject H0” | H0) ≤ 1/T
Note: bounds independent from the prior distribution.
[Zuliani, Platzer, Clarke – HSCC 2010]
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Bounded Linear Temporal Logic (BLTL): Extension of LTL with time bounds on temporal operators.
Let σ = (s0, t0), (s1, t1), . . . be an execution of the model
along states s0, s1, . . .
the system stays in state si for time ti
divergence of time: Σi ti diverges (i.e., non-zeno)
σi: Execution trace starting at state i.
A model for BioNetGen simulation traces
Bounded Linear Temporal Logic
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The semantics of BLTL for a trace σk:
σk ap iff atomic proposition ap true in state sk
σk Φ1 v Φ2 iff σk Φ1 or σk Φ2
σk ¬Φ iff σk Φ does not hold
σk Φ1 Ut Φ2 iff there exists natural i such that
1) σk+i Φ2
2) Σj<i tk+j ≤ t
3) for each 0 ≤ j < i, σk+j Φ1
“within time t, Φ2 will be true and Φ1 will hold until then”
In particular, Ft Φ = true Ut Φ, Gt Φ = ¬Ft ¬Φ
Semantics of BLTL
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Simulations
Oscillations of NFkB and IKK in response to HMGB1 release: ODE vs stochastic simulation
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Verification
Coding oscillations of NFkB in temporal logic
Let R be the fraction of NFkB molecules in the nucleus
We model checked the formula
P≥0.9 Ft (R ≥ 0.65 & Ft (R < 0.2 & Ft (R ≥ 0.2 & Ft (R <0.2))))
The formula codes four changes in the value of R, which must happen in consecutive time intervals of maximum length t
Note: the intervals need not be of the same length
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Verification
Statistical model checking
T=1000, uniform prior, Intel Xeon 3.2GHz
P≥0.9 Ft (R ≥ 0.65 & Ft (R < 0.2 & Ft (R ≥ 0.2 & Ft (R <0.2))))
HMGB1 t (min) Samples Result Time (s)
102 45 13 False 76.77
102 60 22 True 111.76
102 75 104 True 728.65
105 30 4 False 5.76
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Verification
HMGB1 can activate PI3K, RAS and AKT in large quantities
Let PI3Kr, RASr, and IKKr be the fraction of activated molecules of PI3K, RAS, and IKK, respectively
We model checked the formula:
P≥0.9 Ft G180 (PI3Kr > 0.9 & RASr > 0.8 & IKKr > 0.6 )
t (min) samples result time (s)
90 9 False 21.27
110 38 True 362.19
120 22 True 214.38
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Conclusions
Computational modeling is feasible for large models
Temporal logic can be used to express behavioral properties
Statistical Model Checking allows efficient and automatic verification of behavioral properties
Modeling compares qualitatively well with experiments
Further work: parameter estimation importance sampling multi-scale systems
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Acknowledgments
This work supported by the NSF Expeditions in Computing program
Thanks to Michael T. Lotze (University of Pittsburgh) for calling our attention to HMGB1
Thanks to Marco E. Bianchi (Università San Raffaele) for discussions on HMGB1
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Backup slides
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The Cell Cycle
G0: resting, non-proliferating state
G1: cell is active and continuously growing, but no DNA replication
S (synthesis): DNA replication
G2: continue cell growth and synthesize proteins
M (mitosis): cell divides into two cells
The Biology of Cancer. R. A. Weinberg, 2006.
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Bayesian Statistics
Three ingredients:
1. Prior probability Models our initial (a priori) uncertainty/belief about
parameters (what is Prob(p ≥ θ) ?)
1. Likelihood function Describes the distribution of data (e.g., a sequence of
heads/tails), given a specific parameter value
1. Bayes Theorem Revises uncertainty upon experimental data - compute
Prob(p ≥ θ | data)
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Sequential Bayesian Statistical MC
Model Checking
Suppose satisfies with (unknown) probability p p is given by a random variable (defined on [0,1]) with density g
g represents the prior belief that satisfies
Generate independent and identically distributed (iid) sample traces.
xi: the ith sample trace satisfies
xi = 1 iff
xi = 0 iff
Then, xi will be a Bernoulli trial with conditional density (likelihood function)
f(xi|u) = uxi(1 − u)1-xi
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Definition: Bayes Factor of sample X and hypotheses H0, H1 is
prior g is Beta of parameters α>0, β>0
joint (conditional) density of independent samples
Computing the Bayes Factor - I
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Proposition
The Bayes factor of H0:M╞═ P≥θ (Φ) vs H1:M╞═ P<θ (Φ) for n Bernoulli samples (with x≤n successes) and prior Beta(α,β)
where F(∙,∙)(∙) is the Beta distribution function.
No need of integration when computing the Bayes factor
Computing the Bayes Factor - II